The numerical integration schemes for MD and PIMD were based on the reversible
reference system propagation algorithm (RESPA) as implemented in the
PIMD software package.[1]}
The temperature was strongly controlled by attaching massive Nosé-Hoover
chain (MNHC) thermostats[2]}, [3]}, [4]}
to each degree of freedom in the MD simulations.
The MNHC thermostats were also attached to each normal mode representing
the ring polymer in the PIMD simulations.
The fifth-order Suzuki-Yoshida factorization was used for the numerical
integration of the MNHC thermostats.
The simulations were run for \(12.5-25.0\)  ps with the step size of \(0.25-0.43\)  fs
depending on the system and the interatomic potential.
The restraints with a force constant \(\kappa = 4-10\)  hartree were applied at
15 points in the range \(0.16 \le n_i^{} \le 0.98\) .
This resulted in a fluctuation of CN within the order of 0.01.
To deal with the fast oscillation caused by the restraints, the restraint force
was updated 5 times per MD/PIMD step in the RESPA technique.
